共查询到20条相似文献,搜索用时 31 毫秒
1.
Ana Hurtado 《Differential Geometry and its Applications》2008,26(3):227-243
The aim of this paper is to study the stability of the characteristic vector field of a compact K-contact manifold with respect to the energy and volume functionals when we consider on the manifold a two-parameter variation of the metric. First of all, we multiply the metric in the direction of the characteristic vector field by a constant and then we change the metric by homotheties. We will study to what extent the results obtained in [V. Borrelli, Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 30 (2004) 283-292. Erratum on the article: Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 32 (2006) 179-180] for Sasakian manifolds are valid for a general K-contact manifold. Finally, as an example, we will study the stability of Hopf vector fields on Berger spheres when we consider homotheties of Berger metrics. 相似文献
2.
Vladimir S. Matveev 《Topology》2003,42(6):1371-1395
We prove that if two Riemannian metrics have the same geodesics on a closed three-dimensional manifold which is homeomorphic neither to a lens space nor to a Seifert manifold with zero Euler number, then the metrics are proportional. 相似文献
3.
We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T
1
M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary
and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained
results generalize classical theorems on the standard contact metric structure of T
1
M.
Author supported by funds of the University of Lecce. 相似文献
4.
I. G. Nikolaev 《Commentarii Mathematici Helvetici》1995,70(1):210-234
Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural
to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant
curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of
this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL
1-small integral anisotropy haveL
p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that
of constant curvature in theW
p
2
-norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability
results are based on the generalization of Schur' theorem to metric spaces. 相似文献
5.
We consider a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel. With some additional integrability conditions it is called (almost) -manifold and is a natural generalization of the (almost) contact metric and the Sasakian manifolds. There are presented various
methods of constructing examples of such manifolds. There are used structures on the principal bundles and the pull-back bundles.
Then there are considered relations between (almost) -manifolds and transverse almost Hermitian structures on the foliated manifolds.
Research supported by the Italian MIUR 60% and GNSAGA. 相似文献
6.
Vladimir Sharafutdinov Michal Skokan Gunther Uhlmann 《Journal of Geometric Analysis》2005,15(3):499-542
We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over
geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L2-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies
that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit
procedure for recovering the wave front set. 相似文献
7.
On the basis of the so-called phase completion the notion of vertical, horizontal and complete objects is defined in the tangent
bundles over Finslerian and Riemannian manifold. Such a tangent bundle is made into a manifold of almost Kaehlerian structure
by endowing it with Sasakian metric. The components of curvature tensors with respect to the adapted frame are presented.
This having been done it is shown possible to study the differential geometry of Finslerian spaces by dealing with that of
their own tangent bundles.
This work was supported by National Research Coundil of Canada A-4037 (1960–70).
Entrata in Redazione l'8 marzo 1970. 相似文献
8.
Zdeněk Dušek 《Mathematische Nachrichten》2015,288(8-9):872-876
In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo‐Riemannian manifold admits a homogeneous geodesic through arbitrary point. In the present paper this affine method is refined and adapted to the pseudo‐Riemannian case. Using this method and elementary topology it is proved that any homogeneous Lorentzian manifold of even dimension admits a light‐like homogeneous geodesic. The method is illustrated in detail with an example of the Lie group of dimension 3 with an invariant metric, which does not admit any light‐like homogeneous geodesic. 相似文献
9.
Considering Riemannian submersions, we find necessary and sufficient conditions for when sub‐Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. We describe a canonical extension of the sub‐Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results. 相似文献
10.
Vladimir S. Matveev 《Inventiones Mathematicae》2003,151(3):579-609
We show that if all geodesics of two non-proportional metrics on a closed manifold coincide (as unparameterized curves), then
the manifold has a finite fundamental group or admits a local-product structure. This implies that, if the manifold admits
a metric of negative sectional curvature, then two metrics on the manifold have the same geodesics if and only if they are
proportional.
Oblatum 18-IV-2002 & 12-VIII-2002?Published online: 18 December 2002 相似文献
11.
The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the
tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact,
it is found that the only conformal vector fields are Killing vector fields. 相似文献
12.
Christian Pries 《Geometric And Functional Analysis》2009,18(5):1774-1785
Given a C
∞ Riemannian metric g on
P
2 we prove that (, g) has constant curvature iff all geodesics are closed. Therefore is the first non-trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are
closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large
set of C
∞ metrics whose geodesics are all closed and have the same period 2π (called Zoll metrics), but no metric of this set can be
obtained from another metric of this set via an isometry and scaling. As a corollary we conclude that all two-dimensional
P-manifolds are SC-manifolds.
Received: April 2007; Revision: September 2007; Accepted: September 2007 相似文献
13.
It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR
3 with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR
3 is more natural for the study of Legendre curves than the Euclidean metric.This work was done while the first author was a visiting scholar at Michigan State University. 相似文献
14.
Dan Mangoubi 《Mathematische Annalen》2008,341(1):1-13
We consider Riemannian metrics compatible with the natural symplectic structure on T
2 × M, where T
2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive
eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is
that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic
question. 相似文献
15.
In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure. 相似文献
16.
Gideon Maschler 《Differential Geometry and its Applications》2011,29(1):85-92
A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (−a/f)∇df+r=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP1 bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form. 相似文献
17.
We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M 3, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold. 相似文献
18.
Brian Clarke 《Annals of Global Analysis and Geometry》2011,39(2):131-163
We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate
the topology on the manifold of metrics induced by the distance function of the L
2 Riemannian metric—so-called because it induces an L
2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L
1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically
with the completion of the L
2 metric. We also give a user-friendly criterion for convergence (with respect to the L
2 metric) in the manifold of metrics. 相似文献
19.
M. Simon 《manuscripta mathematica》2000,101(1):89-114
The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N
n
is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and
where K is some compact set .
Received: 8 March 1999 相似文献
20.
This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics. 相似文献